Tutu, Jada, and Faith eat lunch together. Tutu contributes 9 sausages, Jada What is the first time after midnight at which the hour hand and minute hand How many integers greater than 0 and less than 100, 000 are palindromes? An

4. A hog trading team sells two hogs for 120 each. They sell one of the hogs for

125 of the price they paid for the hog. They sell the other hog for 80 of the price they paid for it. Do they make a profit or a loss overall, and how much, in dollars, is that profit or loss? Answer: Loss of 6. Solution: The price of the first hog was 4 5 120 = 4 × 24 = 96. There was a profit of 120-96=24 on that hog. The price of the second hog was 5 4 120 = 5 × 30 = 150. There was a loss of 150-120=30 on that hog. There was an overall loss of 30 − 24 = 6.

5. A box of 500 balls contains balls numbered 1, 2, 3, . . . 100 in each of five different

colors. Without ever looking at any of the balls, you are to choose balls at random from the box and put them into a bag. If you must be sure that when you finish, the bag contains at least one set of five balls with identical numbers, then what is the smallest number of balls that you can put in the bag? Answer: 401 Solution: If you happen to put four of the balls with each number in the bag, you will have 400 balls but will NOT have any set of five balls with identical numbers in the bag. Therefore, you need to choose at least 401 balls. On the other hand, if you do choose 401 balls, then you have left behind 99 balls and there must be at least one number that has not been left behind All five balls with that number will be in your bag. So 401 balls will do, but no smaller number will do.

6. Tutu, Jada, and Faith eat lunch together. Tutu contributes 9 sausages, Jada

contributes 8 sausages, and the three girls divide the sausages equally. Faith has brought no food, but gives the other two girls 17 wupiupi coins in exchange for her share of the sausages. How many of the coins should Tutu get? Answer: 10 coins go to Tutu and the other 7 coins go to Jada. Solution: Each girl will get 17 3 sausages, so Faith is trading 17 wupiupi for 17 3 sausages, or 3 coins per sausage. Tutu will trade 9 − 17 3 sausages to Faith, or 10 3 sausages and will get 10 coins. Jada will get the other 7 coins in exchange for 8 − 17 3 or 7 3 sausages.

7. What is the first time after midnight at which the hour hand and minute hand

on an ordinary clock face are perpendicular to one another? Express the time in the format Hour, Minute, Second, with your answer rounded to the nearest second. Assume the clock is a 12 hour clock with hands that move at uniform speeds. Answer: 12 hours,16 minutes, 22 seconds Or 0 hours, 16 minutes, 22 seconds Solution: Let m be the number of minutes past midnight. The hour hand will point at the m 60 × 5 minute mark, or m 12 minutes. The minute hand will be at m. The minute hand will be 90 degrees ahead of the hour hand when it is exactly 15 minute marks ahead. Set m = 15 + m 12 . Solving, we get 11m 12 = 15 or m = 12 × 15 11 = 180 11 = 16 4 11 . To put the answer in the form required, we must convert 4 11 minutes to the nearest second. 4 11 minutes = 4 11 × 60 seconds = 240 11 seconds = 21 9 11 seconds rounded to the nearest second is 22 seconds. Therefore, the time is 16 minutes and 22 seconds past midnight. It is acceptable to give the hour as either 0 or 12. 8. How many integers greater than 0 and less than 100, 000 are palindromes? An integer is a palindrome if its digits are the same when read left to right and right to left. For instance, 2134312 and 353 are palindromes; so are 1001, 99, 5, and 1. Reminder: do not count the number 0. Answer: 1098 Solution: Count the number of 1 digit palindromes, the number of 2 digit palin- dromes, and so on. All 9 positive single digits are palindromes. A 2 digit palin- drome is of the form aa, and there are 9 possibilities for a. There are 9 two digit palindromes- 11, 22, 33, etc. Three digit palindromes are of the form aba. The a can- not be 0, so there are 9 possibilities for a. All 10 digits are possibilities for b. There- fore there are 9 × 10 = 90 three digit palindromes. The four digit palindromes are of the form abba, where again a cannot be 0, so there are also 90 four digit palin- dromes. The five digit palindromes are of the form abcba, with 9 possibilities for a and 10 possibilities for each of b and c. There are thus 9 × 10 × 10 = 900 five digit palindromes. In all, there are 9 + 9 + 90 + 90 + 900 = 1098 positive palindromes less than 100, 000.

9. A polynomial P